Most residents require heat and electricity, so an integrated energy system is needed to meet these demands. To make the heating process clean and economic, a house energy system equipped with photothermal, electrothermal, and photovoltaic devices and a GSHP and an EV was designed and is shown in Figure 1. This system can not only bring about clean heating, but also increase the flexibility of the house energy system.

In the proposed system structure, the energy supply was mainly provided by solar accumulators, RPV systems, and the external power grid, while the heat demands included air heating and hot water. For indoor air heating, the GSHP system was adopted, which can generate lots of heat from small amounts of electricity. Besides, the EHS was equipped to realize electricity-to-heat (E2H) conversion and meet the air heating demand. For the hot water, the solar accumulators were connected to the water system to form the SHW system to provide hot water on sunny days preferentially. Additionally, an EWH system was installed to provide hot water when the sun was not shining.

The proposed structure was grid friendly as the energy system was directly connected to the external power grid. As both solar-thermal and solar-electric conversion efficiency are limited in cloudy and rainy weather, it is necessary to purchase electricity from the external power grid. On the contrary, when the RPV output exceeds the demand of electricity and E2H conversion, the house owner can sell the excess electricity to the power grid. Apart from the power interaction, flexibly controlled MEHLs can optimize the operation of the house system and the external power system. With the improvement of intelligence and the information level, the hot water load, heating load, and the EV can be used as flexible loads. When the power supply is tight, the usage time and practical operation power of these loads can be adjusted flexibly, contributing to peak shaving, frequency stability, and other auxiliary services of the power grid.

In this study, we mainly attempt low-carbon energy dispatching. From the structure shown in Figure 1, carbon emissions were mainly generated by the gas combustion from the external power grid. Therefore, the residents should use energy from solar accumulators and RPV systems as much as possible and optimize the operation process of MEHLs to reduce carbon emissions in the house system. This process is usually performed by home energy management systems (EMS). The default in this study was that the green residence was equipped with an EMS, which was responsible for communication, remote control, load monitoring, etc.

A GSHP is a high-efficient E2H device that uses little electricity to provide lots of heat. Generally, the coefficient of performance (COP) is an effective index to measure electricity consumption and heat supply. The physical power model of a GSHP can be depicted as follows (Marmaras et al., 2016): P G S H P t = Q G S H P h t / C O P t (1) where P G S H P t is the GSHP’s practical electric power at time t, Q G S H P h t is the practical thermal power correspondingly, and C O P t represents the coefficient of performance.

Besides the GSHP, thermostatically controlled load is another type of E2H equipment, including EWHs, EHSs, ACs, and so on (Peirelinck et al., 2021; Wang C et al., 2022; Huang et al., 2019). They have resistivity characteristics when heating. The electric power and thermal power of the E2H conversion process should meet a certain proportion, which can be briefly described as: P E H / E W t = η e , h / w Q E L e , h / w t (2) where P E H / E W t is the electric power for heating and hot water at time t, Q E L e , h / w t is the thermal power correspondingly, and η e , h / w is the E2H conversion efficiency. Super/subscript E H and h refer to heating load, E W and w refer to hot water load, and E L represents electric load.

The SHW system directly converts solar energy into heat. The output is mainly affected by the heat collection efficiency, plate area, and solar radiation intensity. Its physical model can be described as (Li et al., 2023): Q S H W s , w t = 100 × η S H W , h S S H W H S H W t (3) where Q S H W s , w t is the thermal power of the SHW system at time t, η S H W , h is the heat collection efficiency of the solar accumulators, S S H W is the area of the accumulators, and H S H W t is the solar radiation intensity at time t.

RPV systems are a type of common distributed power unit, and their power generation can be described as follows (Soto et al., 2006): P P V t = P R P V G P V t G R 1 + K T P V t − T R (4) where P P V t is the electric power output of RPV at time τ , P R P V is the maximum output of RPV under ideal conditions, K is the temperature coefficient, T P V t is the PV panel temperature at time τ , T R represents the reference temperature under ideal conditions, G P V t is the predictive value of light intensity at time t, and G R represents the rated light intensity under ideal conditions.

The objective function for the operation of green residences is minimizing carbon emissions. Generally, carbon emissions are mainly caused by the burning of fossil fuels. In the proposed energy structure, the energy of a green residence is primarily provided by solar. The photoelectrical and photothermal conversion based on solar energy is clean and zero carbon. Therefore, if the house is energy self-sufficient, carbon emission will not occur. However, when the energy from solar cannot meet the energy demand, residents need to purchase electricity from the external power grid. It is supposed that the power from the external power grid is generated by gas turbines, which is the main cause of carbon emissions. Therefore, the carbon emission is strongly dependent on the generation of gas turbines.

Normally, a quadratic function is used to describe the relationship of the operation cost and the power generation of gas turbines (He et al., 2023). Therefore, the cost of carbon emission is also quadratically dependent on the power generation of gas turbines, as the amount of carbon emission is linearly dependent on the consumption of gas. As such, if the residents purchase electricity/power from the power grid, the cost of the carbon emission can be calculated by quadratic functions. In DA and RT stages, it is inevitable that residents will purchase or sell electricity to maintain the power balance. As a result, the objective function should contain two-time scales. Meanwhile, considering that inaccurate power forecasts will affect the calculation of carbon emissions, a stochastic scheduling method based on a scenario tree is adopted to reduce the uncertainty of prediction. Therefore, the objective function can be described as follows: min ∑ t = 1 T ( C D A t + π s ∑ s = 1 N s C R T t , s ) (5) C D A t = sgn P u D A t ( ξ 1 D A ( P u D A ( t ) ) 2 + ξ 2 D A P u D A t + ξ 3 D A ) Δ t (6) C R T t , s = s g n P u R T t , s ξ 1 R T ( Δ P u R T ( t , s ) ) 2 + ξ 2 R T Δ P u R T t , s + ξ 3 R T Δ t (7) P u D A t = P u , G S H P D A t + P u , e D A t (8) P u R T t , s = P u , G S H P R T t , s + P u , e R T t , s (9) where T is the optimization periods, C D A t is the carbon emission cost of the house energy system at time t in the DA stages, π s is the probability of scenario s, C R T t , s is the carbon emission cost of the house energy system for time t for scenarios in the RT stage, N s represents the number of scenarios, P u D A t depicts the scheduled interactive power between the house system and the external power grid at time t in the DA stages, ξ 1 D A , ξ 2 D A , and ξ 3 D A are the coefficients of carbon emission cost in the DA stage, P u R T t , s depicts the scheduled interactive power between the house system and the external power grid at time t for scenario s in the RT stage, ξ 1 R T , ξ 2 R T , and ξ 3 R T are the coefficients of carbon emission cost in the RT stage, P u , G S H P D A t and P u , e D A t are parts of the scheduled power from the external power grid in the DA stages, which are respectively supplied to GSHPs and electrical load at time t , Δ P u , G S H P R T t , s and Δ P u , e R T t , s are parts of the adjustment power from the external power grid in the RT stage, which are respectively applied for GSHP control and electrical load control at time t for scenario s, and Δ t is the time interval of an optimization cycle. Note that sgn (·) is a jump function. When the variable is greater than 0, the function is 1, which means residents buy electricity from the external power grid. When the variable is less than 0, the function is −1, which means residents sell electricity.

As mentioned above, the proposed strategy contributes to the low-carbon operation of the house system by controlling MEHLs. As residents’ air heating, water heating, and EV charging process will change, temperature constraints and EV charging constraints need to be involved, which are as follows:

For the indoor air heating demand, it is necessary to ensure that the temperature is within the range of the user’s satisfaction, considering that an environment that is too hot or too cold is not suitable for living. The temperature of the house should meet the constraint in Eq. 37. For the hot water demand, there is also a requirement that the temperature is tolerable, and this is shown in the constraint (Eq. 38). T R M , s e t − Δ T R M , s e t ≤ T r , D A / R T t ≤ T R M , s e t + Δ T R M , s e t (37) T T K , s e t − Δ T T K , s e t ≤ T w , D A / R T t (38)

EV needs to be charged to the expected SOC before leaving, shown as: η E V E E V ∑ t = 1 T D A / R T P E V D A / R T t Δ t ≥ Δ S O C s e t (39) where Δ T R M , s e t and Δ T T K , s e t depicts the acceptable deviation of the house and the water, respectively, and Δ S O C s e t is the minimum acceptable SOC for the user.

Eqs 1–39 show that the model is non-linear, which presents two difficulties that need to be solved. First, in the GSHP model, the C O P is determined according to heat supply and electricity generation, and it is a time-varying variable that makes Eq. 1 more complex. To solve this, a little simplification is applied. According to the historical heating data of the house, the average COP values in each period are taken as a reference, and in this strategy, the COP values in different periods are set as constant parameters. Meanwhile, to avoid a large deviation of the actual operation caused by the simplification, the optimization variable Q G H S P h is restricted into a certain percentage, as follows: Q G H S P h t Q G H S P , r e f h t ≤ μ (40) where Q G H S P , r e f h t represents the average heating power from historical data and μ is the percentage, which denotes that the heating power before and after optimization cannot exceed this value, otherwise the actual operation deviation will be large.

Second, the relationship between temperature and thermal power in the water heating and air heating process are non-linear, which is shown in Eqs 10, 12, 37 for house heating and Eqs 13, 15, 38 for water heating. It is difficult to solve these temperature constraints with the objective function directly because of the non-linearity. To solve the problem, the predictions for air heating load and water heating load based on historical temperature data are made and used as thermal demands in the optimization periods. Then, the temperature constraints can transfer into approximate thermal power constraints, as follows: Q w D A / R T t ≥ φ w Q d e m a n d w t (41) φ h min Q d e m a n d h t ≤ Q w D A / R T t ≤ φ h max Q d e m a n d h t (42) where Q d e m a n d w t and Q d e m a n d h t are the thermal demands for hot water and air heating, respectively. φ h min and φ h max are the minimum and maximum controllable coefficients for air heating load, respectively, and φ w is the controllable coefficient for the water heating load.

Finally, when the optimized results are obtained based on the modified constraints in Eqs 40–42, the temperature constraints (10), (12), (37), (13), (15), and (38) will be the verification condition for the results. If the results are not feasible, then the coefficients φ h min , φ h max , and φ w need to be adjusted and the optimization problem needs to be resolved again. The entire solving process is depicted in Figure 2.

To verify the advantages of the proposed strategy, two cases, with and without the proposed strategy, were set to make a thorough comparison in terms of external electricity consumption, carbon emission, house system operation, and residents’ comfort. The two cases are shown in Table 1.

The Monte Carlo method was used to simulate the heating process both in the house and in the water tank. Based on the historical temperature data, which are easily visible in the EMS system, the indoor air heating and hot water demands were calculated according to the temperature data. Additionally, the EV charging process was simulated and the charging demand in each period was predicted. The thermal power and electric power demands are shown in Figure 3, in which QH represents the thermal power used for air heating, QW represents thermal power used for water heating, and EV represents the charging power.

The curves of solar thermal output used for SHW and PV, as well as the common load in the DA stages, are depicted in Figure 4. In a shorter time scale, random noises were also added to simulate the uncertainty from DA and RT prediction.

The proposed strategy involves various parameters, such as carbon emission cost coefficients, MEHL rated values, and controllable coefficients and initialization of MEHLs. Their settings can be seen in Table 2.

As the carbon emission of a single house is relatively small, the proposed strategy rarely generates a profit. However, in the case of the large-scale promotion of MEHLs, the benefits are very optimistic. In this section, 100,000 distributed houses with MEHLs were set and aggregated into a flexible and controllable group to show the benefits afforded by large-scale promotion. Table 3 shows that the proposed strategy can bring large economic benefits in terms of carbon emission cost.

By comparing the carbon emission cost in Case 1 and Case 2, the proposed strategy can save up to 33.07% of carbon emission cost, whether it be a single house or a house group. As such, the proposed strategy can greatly reduce the carbon emissions of the house system with MEHLs and promote clean and low-carbon operation.

The carbon emission cost distribution in the DA periods is described in Figure 5. As observed in Case 1, the highest carbon emission occurred from 8 a.m. to 2 p.m., as did the highest electricity purchase, as EV charging demand is large and relatively concentrated during this period. Normally, the EV starts to charge upon arrival and the charging process is undisturbed. Therefore, in such concentrated charging periods, the house system needs to buy large amounts of electricity from external thermal power plants, resulting in a concentrated distribution of high carbon emission cost. However, in our proposed strategy, i.e., in Case 2, from 8 a.m. to 2 p.m., the electricity purchase and the carbon emission cost were reduced after optimization. Although from 2 p.m. to 7 p.m., the carbon emission cost was higher than that in Case 1, the cost of the full optimization period is obviously lower, which is a great advantage of our strategy.

In other periods, i.e., from midnight to 7.30 a.m. and from 7 p.m. to midnight, the output of the RPV system was relatively low, and most of the energy for hot water and air heating was obtained from the external power grid. To reduce the carbon emission cost of the system, hot water load and heating load were downregulated slightly in Case 2. This means that the proposed strategy can use MEHLs for a lower-carbon optimization and reduce carbon emission costs.

Correspondingly, Figure 6 depicts the incremental carbon emission cost distribution in RT periods compared with the costs of the DA periods. The incremental costs were mainly caused by the prediction errors of photoelectricity, photothermal energy, and residential load. Figure 6 shows that, when there were deviations and uncertainties in RT periods, the carbon emission cost distribution fluctuated less and was more stable in Case 2. In our strategy, the fluctuating outputs of the RPV system and photothermal energy were considered, and MEHLs in the house could match such outputs through flexible control. However, in Case 1, these deviations between the DA and RT periods could only be eliminated through external power purchases. Therefore, the fluctuation range of carbon emission costs at different times was large and the total emission cost was high.

The DA dispatching results of MEHLs in the house system in two cases are compared in Figure 7. The top two panels show that, in Case 2, the operation power of the GSHP is higher than that in Case 1 from 1 p.m. to 4 p.m., while the operation power of the EHS is lower. In the proposed strategy, air heating demand was met by the GSHP, and the electric power of the GSHP was downregulated during most of the optimization periods due to the flexible and moderate control of indoor heating load.

Besides the air heating load supplied by the GSHP and EHS, control of the MHELs in the proposed strategy was also applied to the hot water load and EV. For hot water, shown in the third figure panel, when the photothermal energy was low, the heat supply for hot water load was moderately downregulated. Thus, the EWH, as the supplier in low-photothermal conditions, consumed less electric power, whereas in high-photothermal conditions, the hot water load was met by the SHW system rather than the EWH in our strategy. For the EV, shown in the figure panel at the bottom, the charging power from 8 a.m. to 12 p.m. was downregulated while the charging time was extended and the charging power after 12 p.m. was upregulated to consume more photoelectricity.

In the RT periods, the dispatching and operation of the MEHLs showed a similar trend as that in the DA periods, as shown in Figure 8. Compared with Case 1, the main changes in Case 2 can be summarized with respect to the following three aspects. First, the power of EHS was reduced and the GSHP was fully used to meet air heating demand within adjustable ranges. Second, photothermal energy was fully utilized at noon to increase hot water temperature, and then the heat supply for water in other periods was adjusted within acceptable ranges. Finally, the EV was charged with more flexible power during the entirety of the high-photoelectric periods.

The proposed strategy could meet the electric and thermal demand shown in Figure 3. As shown in Figures 7, 8 above, EV charging was not concentrated between 8 a.m. and 12 p.m. but was completed within a longer time period. Figure 9 depicts the heat supply for air heating and hot water in our strategy. Figure 9 shows that the heat supply for air heating during each period was downregulated, but the total supply was within an acceptable range. The heat supply for hot water in high-photothermal periods exceeded the original demand, and in these periods, more hot water was stored in the tank to maintain a higher temperature over a longer time scale in case the heat supply was downregulated in low-photothermal periods. Therefore, although the energy supply was adjusted after MEHL control, residents’ energy demand could still be met.

Actually, residents’ comfort depends on the temperature when it comes to hot water and air heating load, and also depends on the EV’s SOC when leaving. Therefore, the temperature of hot water, indoor air, and the final SOC were the comfort indexes and were compared in two cases, which are shown in Figure 10. As shown in uppermost figure panel, the indoor temperature rose relatively slowly compared with that in Case 1, but it stayed within the threshold range of deviation acceptable to residents. Meanwhile, in the middle panel, the water temperature fluctuated around the lower limit of deviation acceptable to residents before 8 a.m. then rose to the preset range and dropped at night; these fluctuations are strongly influenced by photothermal energy during the full optimization cycle. Finally, the bottom figure shows that, although the charging time increased, the SOC rose to 100% before the user left. Therefore, the user’s travel demand could be met and no discomfort would generate.

Therefore, the proposed low-carbon-oriented MEHL coordinated control strategy not only has strong advantages in terms of carbon emission costs but also ensures living and traveling standards that residents are satisfied with. It shows reliable supportability in the supply of hot water, indoor air heating, and EV charging when MEHLs are controlled flexibly.